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In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...
In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In abstract algebra, a domain is the noncommutative analogue of an integral domain. ...
Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subrings of fields. Additionally, a commutative ring R is an integral domain if and only if for every element r of the ring, the R-module map induced by multiplication by r is injective (such r are called regular). In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...
In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...
In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
Viewing the underlying commutative ring as a preadditive category, the above criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism, by making use of the bilinear structure on the set of morphisms). A preadditive category is a category that is enriched over the monoidal category of abelian groups. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...
In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ...
The condition 0 ≠ 1 only serves to exclude the trivial ring {0} with a single element. A trivial ring is a ring defined on a singleton set, {x}. The ring operations (* and +) are trivial: x * x = x x + x = x Clearly this ring is commutative. ...
Examples The prototypical example is the ring Z of all integers.
Every field is an integral domain. Conversely, every Artinian integral domain is a field. In particular, the only finite integral domains are the finite fields. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. ...
In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...
Rings of polynomials are integral domains if the coefficients come from an integral domain. For instance, the ring Z[X] of all polynomials in one variable with integer coefficients is an integral domain; so is the ring R[X,Y] of all polynomials in two variables with real coefficients. In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
The set of all real numbers of the form a + b√2 with a and b integers is a subring of R and hence an integral domain. A similar example is given by the complex numbers of the form a + bi with a and b integers (the Gaussian integers). In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = −1. ...
A Gaussian integer is a complex number whose real and imaginary part are both integers. ...
The p-adic integers. The p-adic number systems were first described by Kurt Hensel in 1897. ...
If U is a connected open subset of the complex number plane C, then the ring H(U) consisting of all holomorphic functions f : U -> C is an integral domain. The same is true for rings of analytical functions on connected open subsets of analytical manifolds. In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i represents the imaginary unit, i2 = −1. ...
Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ...
If R is a commutative ring and P is an ideal in R, then the factor ring R/P is an integral domain if and only if P is a prime ideal. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...
In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...
Divisibility, prime and irreducible elements If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
If a divides b and b divides c, then a divides c. If a divides b, then a divides every multiple of b. If a divides two elements, then a also divides their sum and difference.
The elements which divide 1 are called the units of R; these are precisely the invertible elements in R. Units divide all other elements.
If a divides b and b divides a, then we say a and b are associated elements. a and b are associated if and only if there exists a unit u such that au = b.
If q is a non-unit, we say that q is an irreducible element if q cannot be written as a product of two non-units.
If p is a non-zero non-unit, we say that p is a prime element if, whenever p divides a product ab, then p divides a or b.
This generalizes the ordinary definition of prime number in the ring Z, except that it allows for negative prime elements. If p is a prime element, then the principal ideal (p) generated by p is a prime ideal. Every prime element is irreducible (here, for the first time, we need R to be an integral domain), but the converse is not true in all integral domains (it is true in unique factorization domains, however). In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct positive divisors, one and itself. ...
In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...
In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. ...
Field of fractions If R is a given integral domain, the smallest field Quot(R) containing R as a subring is uniquely determined up to isomorphism and is called the field of fractions or quotient field of R. It consists of all fractions a/b with a and b in R and b ≠ 0, modulo an appropriate equivalence relation. The field of fractions of the integers is the field of rational numbers. The field of fractions of a field is isomorphic to the field itself. In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...
Algebraic geometry In algebraic geometry, integral domains correspond to irreducible varieties. They have a unique generic point, given by the zero ideal. Integral domains are also characterized by the condition that they are reduced and irreducible. The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal ideal of a reduced and irreducible ring is the zero ideal, hence such rings are integral domains. The converse is clear: No integral domain can have nilpotent elements, and the zero ideal is the unique minimal ideal. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
In mathematics, the term irreducible is used in several ways. ...
In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
In mathematics, in the fields of general topology and particularly of algebraic geometry, a generic point P of a topological space X is a point such that every point Q of X is a specialization of P, in the sense of the specialization order (or pre-order). ...
In ring theory, a ring R is said to be reduced if it has no non-zero nilpotent elements. ...
Characteristic and homomorphisms The characteristic of every integral domain is either zero or a prime number. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct positive divisors, one and itself. ...
If R is an integral domain with prime characteristic p, then f(x) = x p defines an injective ring homomorphism f : R → R, the Frobenius endomorphism. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...
In commutative algebra and field theory, which are branches of mathematics, the Frobenius endomorphism is a special endomorphism of rings with prime characteristic, a class importantly including fields. ...
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